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164 VIII DIFFERENTIAL CALCULUS

(8,6.4) Let (g_a) be a sequence of mappings of an interval I <=. R into F, and
suppose that, for each n, #„(£) is the derivative of a continuous function f_nexcept for the points £ of a denumerable subset D_n c= L Suppose in addition
that: (1) there exists a point £₀ e I such that the sequence (/_B({₀)) converges
in F; (2) for every point C e I, there is a neighborhood B(Q with respect to I
such that in B(Q the sequence (g_n) converges uniformly. Then for each C e A,
/Ae sequence (/„) converges uniformly in B(Q; O/K/ i/ we jwf /„(£) =

andg(Q = lim #„(£), *Ae/i ^ coery point of A not in \J D_n, /'(£) =
The proof repeats that of (8.6.3), using (8.5.2) instead of (8.5.4).

(8.6.3) yields in particular:

(8.6.5) Let A be an open connected subset in a Banach space, (u_n) a sequence
of differentiate mappings of A into a Banach space F. If for every aeA,
there is a ball B(#) of center a contained in A and such that the series (u^) is
uniformly convergent in B(#), and if there exists a point x₀e A such that
the series (u_n(x₀)) is convergent, then for each aeA, the series (u_n) is uniformly

convergent in B(<z), and its sum s(x) has a derivative equal to ]T u'_n(x) at
every xe A. "⁼⁰

PROBLEMS

1. Let /, g be two real valued differentiable functions defined in an open interval I c: R.
It is supposed that f(t) > 0,#0) > 0,/^x(/) > 0 and g'(t) > 0 in I. Show that if the
function fig' is strictly increasing in I, either fig is strictly increasing in I, or there
exists eel such that fig is strictly decreasing for t ^ c and strictly increasing for
t>c. (Prove that if f(s)lg'(s) <f(s)lg(s\ then for any / < s, f'(t)/ff'(t) <f(t)/g(t).)
Apply to the function

tan t tan a

t tan t — a tan a
in the interval ]a, 77/2 [.

2. (a) Let I be an open interval in R, x₀ e R one of its extremities, /a continuous mapping
of I into a Banach space E. Suppose there is a denumerable subset D of I such that at
each point of I — D,/has a derivative on the right. In order that//(/) have a limit when
/ tends to *₀ in I - D, a necessary and sufficient condition is that (/(/) -f(s))/(t — s)
have a limit when the pair (s, r) tends to (jt₀, *₀) in the set defined by ,9 e I, t e I, s ^ t.
Both limits are then the same; if c is their common value, show that/( /) has a limit in E
when / tends to X_Q in I, and that if/is extended by continuity to I u {jc₀} (3.15.5),
(/(') -/(*o))/(f - *o) tends to c when / tends to x₀ in I. (Use the mean value theorem
and Cauchy's criterion.)



	164 VIII DIFFERENTIAL CALCULUS (8,6.4) Let (g_a) be a sequence of mappings of an interval I <=. R into F, and suppose that, for each n, #„(£) is the derivative of a continuous function f_nexcept for the points £ of a denumerable subset D_n c= L Suppose in addition that: (1) there exists a point £₀ e I such that the sequence (/_B({₀)) converges in F; (2) for every point C e I, there is a neighborhood B(Q with respect to I such that in B(Q the sequence (g_n) converges uniformly. Then for each C e A, /Ae sequence (/„) converges uniformly in B(Q; O/K/ i/ we jwf /„(£) = andg(Q = lim #„(£), Ae/i ^ coery point of A not in \J* D_n, /'(£) = The proof repeats that of (8.6.3), using (8.5.2) instead of (8.5.4). (8.6.3) yields in particular:

	(8.6.5) Let A be an open connected subset in a Banach space, (u_n) a sequence of differentiate mappings of A into a Banach space F. If for every aeA, there is a ball B(#) of center a contained in A and such that the series (u^) is uniformly convergent in B(#), and if there exists a point x₀e A such that the series (u_n(x₀)) is convergent, then for each aeA, the series (u_n) is uniformly 00 convergent in B(<z), and its sum s(x) has a derivative equal to ]T u'_n(x) at every xe A. "⁼⁰

	PROBLEMS 1. Let /, g be two real valued differentiable functions defined in an open interval I c: R. It is supposed that f(t) > 0,#0) > 0,/^x(/) > 0 and g'(t) > 0 in I. Show that if the function fig' is strictly increasing in I, either fig is strictly increasing in I, or there exists eel such that fig is strictly decreasing for t ^ c and strictly increasing for t>c. (Prove that if f(s)lg'(s) <f(s)lg(s\ then for any / < s, f'(t)/ff'(t) <f(t)/g(t).) Apply to the function tan t tan a

	t tan t — a tan a in the interval ]a, 77/2 [. 2. (a) Let I be an open interval in R, x₀ e R one of its extremities, /a continuous mapping of I into a Banach space E. Suppose there is a denumerable subset D of I such that at each point of I — D,/has a derivative on the right. In order that//(/) have a limit when / tends to ₀ in I - D, a necessary and sufficient condition is that (/(/) -f(s))/(t — s)* have a limit when the pair (s, r) tends to (jt₀, ₀) in the set defined by ,9 e I, t e* I, s ^ t. Both limits are then the same; if c is their common value, show that/( /) has a limit in E when / tends to X_Q in I, and that if/is extended by continuity to I u {jc₀} (3.15.5), (/(') -/(o))/(f - o) tends to c when / tends to x₀ in I. (Use the mean value theorem and Cauchy's criterion.)